Certainly! Here’s the translation:

"2. Given that one vertex of a rectangle is at (2,1), another vertex is at (6,5), and the length of the diagonal is 4 units, find the coordinates of the remaining two vertices of the rectangle."

To solve this problem, we start by understanding the given information:

1. Two vertices of the rectangle are at points \( A(2, 1) \) and \( C(6, 5) \).
2. The diagonal length of the rectangle is 4 units.
3. We need to find the coordinates of the remaining two vertices, say \( B(x_1, y_1) \) and \( D(x_2, y_2) \).

### Step-by-Step Solution

1. **Diagonal Length Calculation**:
   The length of the diagonal of a rectangle can be found using the distance formula. Given that \( A(2, 1) \) and \( C(6, 5) \) are two opposite vertices of the rectangle, let's calculate the diagonal length:

   \[
   \text{Diagonal length} = \sqrt{(6 - 2)^2 + (5 - 1)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
   \]

   Since the length of diagonal \( AC \) is \( 4\sqrt{2} \), but it is given that the diagonal length is 4 units, \( AC \) cannot be the diagonal. Thus, \( A(2, 1) \) and \( C(6, 5) \) must be adjacent vertices.

2. **Finding Midpoint and Using Diagonal Properties**:
   For the two unknown vertices \( B \) and \( D \), let's denote:
   
   \[
   B(x_1, y_1) \quad \text{and} \quad D(x_2, y_2)
   \]

   Since the diagonals of a rectangle bisect each other, the midpoint of diagonal \( AC \) is also the midpoint of diagonal \( BD \). Let's find the midpoint \( M \) of \( AC \):

   \[
   M = \left( \frac{2 + 6}{2}, \frac{1 + 5}{2} \right) = (4, 3)
   \]

   This means that the midpoint of \( BD \) is also \( (4, 3) \).

3. **Using Distance to Find Remaining Vertices**:
   We know that the diagonal length \( BD = 4 \). Let’s use the distance formula to express the coordinates of points \( B(x_1, y_1) \) and \( D(x_2, y_2) \) in terms of their distance to the midpoint \( M(4, 3) \).

   \[
   \sqrt{(x_1 - 4)^2 + (y_1 - 3)^2} = 2 \quad \text{and} \quad \sqrt{(x_2 - 4)^2 + (y_2 - 3)^2} = 2
   \]

   Squaring both sides:

   \[
   (x_1 - 4)^2 + (y_1 - 3)^2 = 4 \quad \text{and} \quad (x_2 - 4)^2 + (y_2 - 3)^2 = 4
   \]

   These equations represent two circles of radius 2 centered at \( M(4, 3) \).

4. **Determine Specific Coordinates**:
   Since \( B \) and \( D \) are symmetric about point \( M(4, 3) \), we can express the solutions as:

   \[
   x_1 = 4 + a, \; x_2 = 4 - a, \; y_1 = 3 + b, \; y_2 = 3 - b
   \]

   Substituting into the distance formula gives:

   \[
   a^2 + b^2 = 4
   \]

   Choosing \( a = 2, b = 0 \) and \( a = 0, b = 2 \), we find:

   - For \( (a, b) = (2, 0) \), the points are \( B(6, 3) \) and \( D(2, 3) \).
   - For \( (a, b) = (0, 2) \), the points are \( B(4, 5) \) and \( D(4, 1) \).

5. **Conclusion**:
   The coordinates of the remaining vertices are \( B(6, 3) \) and \( D(2, 3) \) or \( B(4, 5) \) and \( D(4, 1) \).

Would you like more details or have any questions?

### Related Questions:
1. How do you find the midpoint of a line segment?
2. What are the properties of diagonals in a rectangle?
3. How do you determine if two points are adjacent or opposite vertices of a rectangle?
4. Can you explain the distance formula and its applications in geometry?
5. How do you solve for unknown variables using geometric constraints?

### Tip:
When working with geometric shapes, visualizing the problem by sketching can often make it easier to understand and solve.

Given that one vertex of a rectangle is at (2,1), another vertex is at (6,5), and the length of the diagonal is 4 units, find the coordinates of the remaining two vertices of the rectangle.

Explore how to find the missing vertices of a rectangle when given one pair of vertices and the diagonal length. This problem involves coordinate geometry, the distance formula, and the properties of diagonals in a rectangle. Ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation